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Theorem ltrncnvatb 30872
Description: The converse of the lattice translation of an atom is an atom. (Contributed by NM, 2-Jun-2012.)
Hypotheses
Ref Expression
ltrnatb.b  |-  B  =  ( Base `  K
)
ltrnatb.a  |-  A  =  ( Atoms `  K )
ltrnatb.h  |-  H  =  ( LHyp `  K
)
ltrnatb.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrncnvatb  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( P  e.  A  <->  ( `' F `  P )  e.  A
) )

Proof of Theorem ltrncnvatb
StepHypRef Expression
1 ltrnatb.b . . . . . 6  |-  B  =  ( Base `  K
)
2 ltrnatb.h . . . . . 6  |-  H  =  ( LHyp `  K
)
3 ltrnatb.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
41, 2, 3ltrn1o 30858 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F : B
-1-1-onto-> B )
543adant3 977 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  F : B
-1-1-onto-> B )
6 simp3 959 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  P  e.  B )
7 f1ocnvdm 6010 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  P  e.  B )  ->  ( `' F `  P )  e.  B
)
85, 6, 7syl2anc 643 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( `' F `  P )  e.  B )
9 ltrnatb.a . . . 4  |-  A  =  ( Atoms `  K )
101, 9, 2, 3ltrnatb 30871 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( `' F `  P )  e.  B
)  ->  ( ( `' F `  P )  e.  A  <->  ( F `  ( `' F `  P ) )  e.  A ) )
118, 10syld3an3 1229 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( ( `' F `  P )  e.  A  <->  ( F `  ( `' F `  P ) )  e.  A ) )
12 f1ocnvfv2 6007 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  P  e.  B )  ->  ( F `  ( `' F `  P ) )  =  P )
135, 6, 12syl2anc 643 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( F `  ( `' F `  P ) )  =  P )
1413eleq1d 2501 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( ( F `  ( `' F `  P )
)  e.  A  <->  P  e.  A ) )
1511, 14bitr2d 246 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( P  e.  A  <->  ( `' F `  P )  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   `'ccnv 4869   -1-1-onto->wf1o 5445   ` cfv 5446   Basecbs 13461   Atomscatm 29998   HLchlt 30085   LHypclh 30718   LTrncltrn 30835
This theorem is referenced by:  ltrncnvat  30875
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-undef 6535  df-riota 6541  df-map 7012  df-plt 14407  df-glb 14424  df-p0 14460  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-hlat 30086  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839
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